Structural Diversity from Anion Order in Heteroanionic Materials

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Article

Structural Diversity from Anion Order in Heteroanionic Materials Nenian Charles, Richard J Saballos, and James M. Rondinelli Chem. Mater., Just Accepted Manuscript • DOI: 10.1021/acs.chemmater.8b01336 • Publication Date (Web): 30 Apr 2018 Downloaded from http://pubs.acs.org on May 1, 2018

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Chemistry of Materials

Structural Diversity from Anion Order in Heteroanionic Materials Nenian Charles,†,‡ Richard J. Saballos,‡ and James M. Rondinelli∗,‡ †Department of Materials Science and Engineering, Drexel University, Philadelphia, Pennsylvania, USA ‡Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois, USA E-mail: [email protected]

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Chemistry of Materials 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract Heteroanionic materials leverage the advantages offered by two different anions coordinating the same or different cations to realize unanticipated or enhanced electronic, optical, and magnetic responses. Beyond chemical variations offered by the anions, the ability to control the anion order present within a single transition metal polyhedron via anion-sublattice engineering offers a potentially transformative strategy in tuning material properties. The set of design rules for realizing and controlling anion order, however, are incomplete, which is due in part to the limited anion-ordered diversity in known structures. This aspect makes formulating such principles from experiment alone challenging. Here, we demonstrate how computational methods at multiple levels of theory are effective at investigating the anion site order dependencies in heteroanionic materials, HAMs, and enable the construction of crystal-chemistry principles. Our approach relies on a database of anion ordered structure variants in which we manipulate the lattice degrees of freedom through the incorporation of structural distortions. Structure-property relationships and anion-order descriptors are data mined from group theoretical techniques and density functional theory calculations. Using our combined computational scheme, we uncover a hybrid improper mechanism to stabilize polar phases, propose the chemical link between local and long ranger anion order, and detail the sequence of order-disorder/displacive transitions observed experimentally in the oxyfluoride Na3 MoO3 F3 . Our method is scalable and transferable to many heteroanionic chemistries and crystal families, facilitating the construction of heteroanionic materials design principles.

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Introduction At present, there is renewed interest in so-called heteroanionic materials (HAMs), which aim to achieve tailored material properties by incorporating multiple anionic species of varying charge, ionic radii, and electronegativity into inorganic crystalline compounds. 1 The utility of anion engineering lies in the potential to combine the advantageous properties of each individual anionic species together or even realize an unanticipated response. This approach has already proved fruitful in the development of water-splitting catalysts, phosphors, and battery cathodes that out perform their homoanionic, for example, oxide-only counterparts. 2–4 In many heteroanionic materials, the ability to control the degree of anion order in the crystal is critical to designing desired properties 5 and has broad technological implications, including high-Tc superconductivity, second harmonic generation (SHG), electrochemical activity, and multiferroicity. 4,6–9 Researchers active in heteroanionic oxynitrides (materials with oxide and nitride anions) and oxyfluorides (oxide and fluoride anions) have begun cataloging anion ordering principles based on observed trends in known compounds. 5,10–12 The anion ordered polyhedra form heteroleptic anionic groups, e.g., [NbOF5 ]2− , 13 that are distributed throughout the crystal. One guideline gleaned from these efforts is that the short range covalent metal–anion interactions give rise to A` Mm (Ox Xy )n compositions in crystal structures defined by polyhedral units with local anion order, e.g., M Ox Xy , where X =N3− , S2− , F1− , etc. For many HAMs, short-range interactions strongly favor anionic groups with site order. The driving force is typically due to the relative difference in electronegativity between the two ligands. 14 For instance, in transition metal (M ) oxyfluoride compounds, stronger covalent M –O interactions result in shorter bond lengths compared to the more ionic M –F interactions. The characteristic differences in bond lengths manifest as M cation displacements towards the oxide ions, particularly for d0 transition metals. 15,16 In structures with [M OF5 ]n− units, the M cation off-center towards the corner of the octahedra, resulting in C4v point symmetry. In d0 [M O2 F4 ]n− units, the different bond length preferences produce oxide ion arrangements that 3



distort the M cation towards an edge, i.e., a cis (C2v ) rather than trans (D2h ) configurations. Similarly, fac configurations are favored over mer for [M O3 F3 ]n− anionic groups. These observations have been independently confirmed by electron scattering, nuclear magnetic resonance, and vibrational spectroscopy experiments as well as computationally. 10,17–22 The preference for cis and fac configurations can be violated if non-d0 transition metals cations are incorporated into the structure; for example, trans configurations have been reported in iodine oxyfluoride salts. 23 Despite the presence of locally ordered anionic groups, many heteroanionic materials are found to be disordered from X-ray diffraction measurements. 22 In these cases, the collective arrangement of [M Ox Xy ]n− polyhedra lack long-range order; i.e., two or more configurations of locally ordered polyhedral units are equally favorable in neighboring unit cells, thus, substitutional disorder ensues. This type of anion disorder in HAM crystals is sometimes referred to as orientational disorder , 18,24 owing to the lack of a unique orientation of the [M Ox Xy ]n− units in the superstructure. Evidence of ordered heteroleptic units giving rise to substitutional disorder has been observed in perovskite oxynitrides. 10 Interestingly, long range anion order, i.e., the correlated arrangement of multi-anion polyhedral units, is less commonly observed and thus poorly understood. In cases where true long-range anion order is absent, electron scattering techniques can be used to characterize the local structure variations by exploiting the small cation distortions in the metal-anion polyhedra that accompany local anion order. 25 Indeed, electron scattering has been successfully used to elucidate the underlying patterns of anion order in FeOF and Na3 MoO3 F3 . 18,26,27 When combined with bond-valence calculations, 28,29 this approach provides a more detailed description of anion order. Although few studies address anion ordering directly, several hypotheses have been proposed to explain the prevalence of long range disorder in heteroanionic compounds, including electrostatic repulsion between adjacent heteroleptic units, cation size effects, and the presence/absence of secondary distortions. 10,18,25,30,31 Nonetheless, there is no consensus

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as to which effect is most important in achieving the desired order across different chemistries. Fuertes attempted to predict anion distributions in heteroanionic materials on the basis of Pauling’s electrostatic valence rule. Analysis of 28 materials and 78 crystallographically independent anion positions showed that when anion ordering takes place, higher charged anions typically occupy sites that give larger electrostatic bond strengths. 32 However, this bond strength-based method only predicts anion distributions at 38% accuracy for oxychlorides and 20–25% accuracy for oxynitrides and oxyfluorides. 32 More recently, Talanov et al. used representation theory to enumerate the structural distortions consistent with anion order in 261 hypothetical structures derived from the P m¯3m lattice of ABX3 perovskites. 12 They found that the theoretically derived structures capture the structural features of experimentally observed compounds. Another major conclusion of their study is that although octahedral rotation and tilt distortions are commonly exhibited in low-symmetry anion ordered structures, they are not a driving force for anion ordering in ABX3 perovskites. 12 Although the authors of Ref. 12 believe their method to be complete, they also describe it as very cumbersome. As a result, one would infer that the representation analysis as carried out in Ref. [ 12 ] likely cannot not be easily extended to different structure types. Here, we employ a bottom-up approach to elucidate the crystal-chemistry principles of anion ordering on the crystallographic scale for heteranionic oxyfluoride materials by combining known ordering principles with multiple levels of theory and computation. The foundation of this study is a derivative structure generating algorithm tailored to build inorganic heteroanionic materials with ordered anionic group by mimicking supramolecular assembly. From this starting point, we data mine structural and chemical trends from group theoretical and density functional theory analyses. It is worth mentioning that similar atomic structure engineering approaches have been successfully applied to investigate ferroelectricity, phase transitions, and disorder in crystalline materials. 33–38 From our crystals chemistry analyses, we extract several salient structural trends including: (i) How structural distortions, such as octahedral rotations, can couple to anion order to drive nonpolar to polar transitions

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through a hybrid improper mechanism; (ii) The avenue to actively controll local anion order, for example cis vs. trans is likely tied to the selection of the cation in the [M OF5 ]n− units; and (iii) For highly ordered oxyfluoride compounds the allowed/accessible space group symmetries of the 3D crystal are strongly directed by the point group symmetry of the local [M OF5 ]n− units. Moreover, we demonstrate that thermodynamic analysis of known ordered and disordered oxyfluoride phases, such as Na3 MoO3 F3 and K2 NaTiOF5 , can identify accessible low temperature variants that enable one to probe the interplay between chemistry and anion order. An interesting auxiliary use of such low energy structures could be for guiding structural solutions of HAMs where correlated disorder dominates.

Enumerating Anion Ordered Derivative Structures We first describe how each component of our workflow contributes to the investigation of the interactions governing anion order in heteroanionic materials (Fig. 1). The derivative structure code is designed to exhaustively sample the configurational space of anion order materials and enumerate unique structures delineated by space group symmetry. Coupling the derivative structure algorithm with group-theoretical methods and density functional theory (DFT) calculations enables us to efficiently probe the contributions of electrostatics, chemical pressure, and structural distortions to the presence/absence of long range anion order in heteroanionic materials. This approach can also be extended to other structural and chemical families including, oxynitrides, oxysulfides, and ordered anion vacancies from which anion ordering principles may be constructed. Generally, derivative structures are derived from a basic/parent structure. 39,40 Derivative structures can be categorized by how there are generated: ‘derived’ by superimposing lattice distortions, substitution of constituent ions, or a combination of both. 39 A derivative structure also shares a group-subgroup relationship with the parent, i.e., the basis vectors and Wyckoff positions of the derivative structure correspond to lattice sites in the parent cell. 40,41 The

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Start

compute electronic and dielectric properties with DFT

parent structure

identify building block unit for tiling (M X6 )

enforce stoichiometry (M Yx X6−x )

decorate lattice

Stop

∆E(i) ≤ τ

update structures

impose displacive distortions (tilts)

∆E(i) > τ

symmetry analyses (sort and classify)

derivative structure generation

evaluate stability

DFT calculations

data analysis

Figure 1: Workflow for anion engineering. Trial structures are obtained through the derivative structure algorithm Site Ordered Crystal CReator (SOCCR, steps to the left of the vertical broken line). We employ data analysis and the superposition of structural distortions to enable structure classification, correlation, and phase stability with electronic structure calculations. group-subgroup relationship also enable us to characterize the connection between the parent and derivative structures using irreducible representations (irreps) in a group theoretical analysis. Please note that the Miller and Love convention is used throughout this manuscript for irrep labeling. 42 For a detailed overview of the group theory methods and strategies employed throughout this manuscript please refer to Refs. [ 35,43–46 ]. Derivative structures are used in computational studies to investigate a broad range of phenomena including chemical ordering in binary alloys, twinning, superlattices formation and vacancy ordering in nonstoichiometric compounds. 40,47–52 As a result, several tools and algorithms have been devised to build derivative structures depending upon the class of materials and properties being investigated. Example derivative structure generation methods include the Ferreira, Wei, and Zunger (FWZ) algorithm, random sampling, particle-swarm optimization algorithms, global space-group optimization, evolutionary algorithms and cluster expansions. 53–58 Although these approaches have enable computational researchers to tackle a variety of problems, they may require significant user involvement when applied to atypical materials such as those with multiple anions. For this reason, we introduce a method tailored

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to HAMs, which generates derivative structures based on the supramolecular assembly, i.e. where superstructures are built up from [M Ox Xy ]n− basic building block units (BBUs). The main steps of the workflow include to: 1. Identify likely parent lattices based on target chemistry and stoichiometry. For example, F m¯3m symmetry with a cubic lattice for the A2 BM X6 double perovskite. 2. Define parent structure (lattice vectors and fractional coordinates of atomic positions). 3. Define coordination sphere to identify anionic BBUs. In the double perovskite parent cell A2 BM X6 , the BBUs are [M X6 ]n− or [AX12 ]m− . 4. Enforce stoichiometry constraints for derivative structures with two or more anions. For the double perovskite, A2 BM X6 → A2 BM Ox X6−x where x = 1, 2, 3. (a) Decorate high-symmetry lattice to generate possible atomic configurations. (b) Classify derivative structures by space group symmetry. (c) Sort derivative structure, removing duplicates, to determine the unique set of derivative structures. 5. Apply structural distortions such as octahedral rotations to reduced set of structures and find symmetry of structure. 6. Data mine structural, energetic, electronic, properties etc. 7. Calculate phase stability and properties using electronic structure methods. A space of metastable materials can be created by searching for structures where the difference in energy from the lowest energy structure (∆E) is less than or equal to an energy threshold (τ ). 8. Terminate the process once the lowest energy structure is identified if the ground state structure of a specific chemistry is desired. 8


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Steps 1–5 are implemented in the SOCCR algorithm available at https://github.com/ MTD-group/MTDGderivative-structures. It is worth mentioning that the current implementation of this approach is designed to enumerate structural variants with a fixed stoichiometry within the structure type defined as the parent. Furthermore, given that the representative structure types may vary considerably depending on the chemistry of heteroanionic compounds being investigated, the current algorithm does not automatically change structure types as in other codes. 57 In this light, we acknowledge that SOCCR is not as mature or sophisticated compared to established derivative structure codes such as ATAT 53 or enumlib. 40,59,60 However, we find that SOCCR’s tailored supramolecular assembly approach reduces the number of trial configurations sampled and guarantees that unique derivative structures possess the [M Ox Xy ]n− motif. Resulting in significant time saving during pre- and post-processing. We note the SOCCR code is capable of handling structures beyond perovskites and chemical orderings other than anions (cation ordering and vacancies). Furthermore, based on our benchmarking (see Supporting Information), we contend that our use of random sampling, parallelization and a simple user interface makes SOCCR an attractive alternative to other codes designed to create derivative structures at fixed compositions.

Structural distortions Prior work on heteroanionic materials has shown the coexistence of anion order and displacive distortions, 5,25,26 suggesting the latter can be used as a practical handle to achieve the desired anion order. Researchers commonly view short-range interactions such as the second-order Jahn-Teller (SOJT) distortion as the driving force for the formation of ordered anionic groups. 14,17,61 On the other hand, long-range cooperative distortions, such as octahedral rotations and tilts, may act to direct the collective arrangement of anionic groups, i.e., promoting or suppressing substitutional disorder in the compound. Evidence of anion directing octahedral rotations have been observed experimentally in oxyfluoride double perovskites, Ruddlesden–Popper phases and oxynitride single perovskite compounds. 5,25,62,63 9



Table 1: Change in symmetry by coupling tilt patterns (displacive modes) to ordering modes. None means coupling is not allowed from the perspective of isotropy group theory. 64 However, the hypothetical structures can still be created using a distortion routine implemented in SOCCR. tilt pattern no anion order X3− Γ− 4 a0 a0 a0 F m¯3m P 4/nmm I4mm a0 a0 c+ P 4/mnc P 421 2 Ama2 P ccm P 4nc 0 + + ¯ ab b P 42 /nnm P 42m P 42 nm P 222 P nn2 + + + ¯ a a a P n3 P 222 P nn2 + + + a b c P nnn P 222 P nn2 a0 a0 c− I4/m C2/m I4 P 4/n Cm a0 b− b− C2/m P 21 /m Cm P ¯1 a− a− a− R¯3 P ¯1 none − − − ¯ ¯ a b c P1 P1 none 0 + − ab c C2/c P 2/c Cc C2 C2 − − + a a c P 21 /c P 21 Pc P2 + + − a a c P 42 /n P ¯1 Pc P 222 P 42 The following structural distortion analysis employs the group theoretical approach as implemented in the ISOTROPY Software Suite. 64 In Table 1, we aggregate the results of the change in space group symmetry due to the superposition of Glazer tilt patterns with the ordering irreps X3− and Γ− 4 for the A2 BM OF5 double perovskite oxyfluoride. Here the order-parameter directions for the X3− and Γ− 4 irrep labels are (a, 0, 0). From this group-theory investigation, we observe that symmetry reductions can be induced by superimposing the centrosymmetric ordering irrep X3− with the a0 a0 c+ , a+ b+ c+ , a0 b+ c− and a− a− c+ tilt patterns to produce chiral and chiral-polar structures. Interestingly, the loss of inversion symmetry in the chiral-polar structures results from the superposition of two centric mode distortions i.e. in-phase rotation, X3+ and out-of-phase tilt, Γ+ 4 . Given that the symmetry reduction to 10


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these chiral-polar structures is not directly induced by a polar mode, the nonpolar to polar transition can be considered hybrid improper. 38 The nature of the coupling can be obtained by considering the symmetry permitted terms in a polynomial expansion of the free energy change ∆F using anion order O, in-phase rotations (R) and out-of-phase tilts (T ) of the anionic units, and the polarization (P ) as order parameters with the parent F m¯3m symmetry serving as the reference structure as,

∆F = α0 O2 + α1 R2 + α2 T 2 + α3 P 2 + β0 O4 + β1 R4 + β2 T 4 + β3 P 4 + γ0 O2 R2 + γ1 O2 T 2 + γ2 O2 P 2 + γ3 R2 T 2 + γ4 R2 P 2 + γ5 T 2 P 2 + δ0 ORT P .

(1)

Note that for F m¯3m, the in-phase rotations (R) are represented by irrep X3+ and the outof-phase tilts (T ) by Γ+ 4 . Although Equation 1 consists only of even-order terms, the last quad-linear term couples the acentric displacements that would produce P linearly to the ordering and cooperative anionic displacements. This form is analogous to how A-site driven polarization occurs in hybrid improper ferroelectrics. 65 Indeed, if the free energy expansion reference is the anion-ordered P 4/nmm structure (a0 a0 a0 ) the invariant polynomial can be rewritten as follows,

∆F = a1 R2 + a2 T 2 + a3 P 2 + b 1 R 4 + b2 T 4 + b3 P 4 + g3 R2 T 2 + g4 R2 P 2 + g5 T 2 P 2 + d0 RT P.

(2)

Note that for P 4/nmm, the in-phase rotations (R) are represented by irrep Γ− 1 and the 11



out-of-phase tilts (T ) by Γ+ 5. Interestingly, an anion ordered P 4/nmm structure has been observed experimentally for the compound KNaNbOF5 . 66 Thus, it is conceivable that one could design anion ordered hybrid improper ferroelectrics by choosing a chemistry which supports both anion ordering 67 and octahedral tilting or alternatively tuning experimental conditions for a known compound such as KNaNbOF5 with epitaxial strain. Condensing independently the acentric ordering irrep, Γ− 4 results in the the polar anion ordered I4mm structure. Table 1 shows that when this anion order exists with octahedral rotations and tilts, it always produces in a polar phase of lower symmetry. This highlights a key distinction among the noncentrosymmetric variants generated from X3− and Γ− 4 anionorder irreps: The Γ− 4 irrep induces a polarization based on the order of anions in the lattice, similar to the ABAB stacking of tetrahedra in the wurtzite structure. Although the coupling of secondary distortions (rotations) could enhance the magnitude of the polarization, the total electric polarization remains irreversible, making it more suitable for piezoelectric and nonlinear optical applications. In contrast, anion order due to X3− with coexisting octahedral rotations would permit a switchable electric polarization albeit through the reversal of the displacive mode rather than by a reconfiguration of the static anion order analogous to improper ferroelectrics. 68 Similar investigations can be performed for other combinations of anion-order irreps with secondary distortions and varying anion ratios.

Data Mining The structural consequences arising from anion order can be rapidly assessed using data mining techniques. To facilitate this process, we analyze the structural variants obtained from our derivative structure algorithm using group theory to identify the irreps, which characterize the symmetry breaking from the F m¯3m space group. This process is possible because the lattice decoration performed by the algorithm is equivalent to splitting the Wyckoff site(s) occupied by the anion (24e) followed by enumerating the unique structures 12


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under the stoichiometric constraint.

Anion order and group theory For all derivative structures investigated, we find that the symmetry lowering induced by anion order is described by modes with different order parameter directions using wavevectors at the Brillouin zone center (Γ) or boundary (X), which is generally consistent with the aforementioned analysis by Talanov et al. 12 All structure variants in the anion-ordered A2 BM Ox F6−x perovskites (x = 1, 2, 3) obtained with SOCCR can be achieved using a single irrep at (Γ or X) or the superposition of the two irreps. Note that in our mode crystallography analysis octahedral rotations and tilts irreps never manifest with an ordering component, i.e., they are only displacive. Order parameters which are only displacive are represented g ) above the irrep symbol. hereafter with a tilde (OP In Fig. 2 we report the frequency with which anion order is driven by in-phase rotations − + + g f+ 69 (X Q3 (Γ+ 3 ), JT Q2 (X2 ), polar (Γ4 ) and 3 ), out-of-phase tilts (Γ4 ), Jahn-Teller (JT)

antipolar (X3− ) displacements as the primary order parameters. We use ‘driven by’ to mean that the mode provides a sufficient symmetry reduction for the accompanying displacement to occur freely. We define primary order parameter(s) as mode(s) that singularly (or coupled) generate the low symmetry distorted structure. (Note that Fig. 2 only considers the effect of the modes listed. Anion order may be induced by other modes not presented.) Similar to the findings in Ref. 12 , we also conclude that in A2 BM Ox F6−x double perovskites the octahedral rotations and tilts by themselves do not generate anion ordered structures. In contrast, the JT, polar and antipolar irreps have both displacive and order characteristics. Interestingly, we find that for x = 1, 2, and 3 investigated, the octahedral rotations and tilts are necessary participants in the symmetry breaking that leads to certain anion ordered phases. Fig. 2 also shows that the most frequently occurring primary order parameters are polar and antipolar displacements in all stoichiometries considered. For the A2 BM OF5 stoichiometry [Fig. 2(a)], we find that the polar (Γ− 4 ) and antipolar

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(a)

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(b)

(c)

Figure 2: Structure-symmetry-sorting diagram for anion-order modes in double perovskites with stoichiometry (a) A2 BM OF5 , (b) A2 BM O2 F4 and (c) A2 BM O3 F3 . Each box at the intersection of a row and column describes the effect of coexisiting rotation, tilt, Jahn-Teller, polar and antipolar modes with each other, whereby the numerals within each box indicate the number of unique symmetry lowering irrep combinations present. Gray squares indicate no coupling (0) and the shade of the colored squares corresponds to the coupling frequency, i.e., deeper shade, higher frequency of occurrence. (X3− ) irreps can independently generate anion ordered structures, space groups I4mm and P 4/nmm respectively. The appearance of polar and antipolar modes are perhaps obvious since they both naturally give rise due to displacements that locally lift inversion symmetry in the [M OF5 ]n− units. In the case of the antipolar mode, the polar displacements in the anion units are globally arranged to be out of phase, for example in KNaNbOF5 . 66 To arrive at all of the other structures generated from the listed modes requires a superposition of the anion-order order parameter with either polar or antipolar displacements. Interestingly, we see that the symmetry breaking required to arrange oxygen and fluorine in a 1:5 ratio can be achieved with the following combinations: polar ⊕ rotation or antipolar ⊕ tilt but not polar ⊕ tilt or antipolar ⊕ rotation. The A2 BM OF5 order can also be obtained by coupling the JT Q2 mode to variants of the polar and antipolar modes. However, the symmetry breaking induced by the JT Q3 does not satisfy the stoichiometric constraint. Perhaps as expected, anion ordering is also driven by the coupling between polar and antipolar displacements. The structure-symmetry-sorting diagram for anion ordered A2 BM O2 F4 differs slightly from that of A2 BM OF5 [Fig.2(b)]. Here, the irreps that independently generate anion ordered structures are the JT Q2 , JT Q3 , and polar distortions: The JT Q2 distortion results in an 14


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anion-ordered structure on a primitive lattice with P 42 /mnm symmetry whereas the JT Q3 distortion produces a body-centered I4/mmm space group. In addition, the two-in–two-out Q2 JT and four-in–two-out Q3 JT displacements naturally yield anionic groups with trans arrangements of oxide ions. On the other hand, the structure obtained from the singular polar order parameter Γ− 4 generates space group Imm2 with cis arrangements of the oxide ions. Last, there are no antipolar modes (X3− ) that independently generate anion ordered arrangements in contrast to the aforementioned A2 BM OF5 case. As a result, anion order relies on the superposition of either (both) polar or (and) antipolar modes with rotations, tilts, and JT Q2 . Interestingly, the coupled irreps responsible for anion order in A2 BM O3 F3 [Fig. 2(c)] are identical to those in A2 BM OF5 . Again, we see that only the polar and antipolar modes alone generate anion-ordered structures, i.e., space group R3m and P ¯43m respectively. The isomeric configuration of the oxide ions in the R3m and P ¯43m structures is fac. It is worth mentioning that structures with mer isomeric configurations do also occur, however, they are not generated by a single ordering irrep at Γ or X. In other words mer configurations always require a superposition of 2 or more irreps. We investigate the isomeric configurations in more detail next.

Geometric isomerism Here, our complete search of A2 BM O2 F4 and A2 BM O3 F3 anion ordered configurations enables us to enumerate structures with cis, trans, fac and mer polyhedral units; the corresponding point group symmetries are mm2, 4/mmm, 3m, and mm2 respectively. Now, we data mine our structural space to further classify the derivative structures based on isomeric configurations of the heteroleptic anionic units. Derivative structures are labeled cis, trans, fac or mer in instances where all of the polyhedral units are alike. Structures where two types of isomer configurations coexist are classified as mixed. We note that arrangement of polyhedral building units in many heteroanionic materials

15



12

60

count

10 50

40

mixed cis trans

8 6 4 2

Pmm2 Pmc21 Pma2 Pmn21 Cmn21 Amm2 Abm2 Ama2 Imm2 Pmmm Pbcm Pmmn Pnma Cmmm

0

count

30

3

I4/mmm

P42/mnm

P42/mcm

space group symmetry

P4/mmm

0

_

_

0

1

P4

10

2

P42

count

20

P1 P2 P21 C2 Pm Pc Cm P2/m P21/m C2/m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Figure 3: Occurrence of A2 BM O2 F4 derivative structures with cis, trans, and mixed isomer configurations sorted by space group symmetry. resembles the assembly imposed symmetry constraints found in supramolecular chemistry. Each polyhedral unit has a defined point symmetry, which must be compatible with the point symmetry in the 3D arrangement of units in the superstructure. To that end, establishing correlations between the local anion configurations and the space group symmetry of possible low temperature phases provides useful structural information that may be utilized in subsequent crystal engineering of heteroanionic materials. Fig. 3 depicts the distribution of cis, trans, and mixed structures by space group symmetry for A2 BM O2 F4 . We see that the isomer configuration with the highest point group symmetry 4/mmm (trans) gives rise to the derivative structure with the highest space group symmetry, I4/mmm. Fig. 3 also reveals that the trans configuration occurs less than cis in heteranionic double perovskites based on enumerating all possible structure derivatives. The presence of multiple isomeric anionic units acts to reduce the space group symmetry (see space groups

16


Page 17 of 35

15

count

100 mixed fac mer

10

Amm2 Fmm2 Pmma Pmmn Cmcm Cmma

50

Pma2 Pca21 Pmn21 Pna21

0

Pmm2 Pmc21

5

75

count

10

_

P43m

P213

R3

R3m

_

P421m

P42nm

0

P42cm

5

P42/m

_

0

count

25

P1 P2 P21 C2 Pm Pc Cm Cc P2/m P21/m P2/c P21/c

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



Figure 4: Occurrence of A2 BM O3 F3 derivative structures with f ac, mer and mixed isomer configurations sorted by space group symmetry. corresponding to mixed in Fig. 3). The orthorhombic symmetry Cmmm (no. 65) is the highest space group with a mixed structure. Indeed, we see that the occurrence of mixed structures increases as the number of symmetry operations in the crystal decreases; the most common structure exhibits monoclinic symmetry in space group P m (no. 6). Fig. 4 shows the isomer configuration data for A2 BM O3 F3 double perovskites. Like the A2 BM O2 F4 structures we find that polyhedral unit with higher point group symmetry (fac, 3m) leads to the most symmetric superstructures. Interestingly, we find that the 3m symmetry of the fac unit when assembled is compatible with cubic symmetries, P ¯43m (no. 215) and P 21 3 (no. 198). Indeed, the fac configuration is also compatible with the trigonal space groups R3m (no. 160) and R3 (no. 146). The lower symmetry (mm2) mer units are not observed in either of these crystal systems. The first structures with mer isomers are tetragonal with space group P ¯421 m (no. 113). Interestingly, although the fac configuration

17



30 25 20 15 10

P4/nmm

I4mm

P4mm

P42/m

Cmcm

Pmn21

Pma2

Pmc21

P21/m

0

Cm

5 Pm

energy (meV/atom)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Figure 5: Total energies for 17 structurally optimized anion ordered variants enumerated for K2 NaTiOF5 calculated at the DFT-PBEsol level. is energetically favored over mer for d0 cations, 19 mer (19.2%) ordered derivative structures occur with higher frequency compared to fac (11.6%). Structures with multiple isomeric anionic units (mixed) characteristics make up 69.2% of the unique structures determined. For the A2 BM O3 F3 case, mixed structures first appear in the trigonal space group R3m (no. 160), but occur most frequently in the monoclinic structures, especially P m (no. 6).

Structure Stability One way to explore the nature and degree of anion order relies on examining the lattice energy of various derivative structures. Ionic models have been used to investigate anion order in silicon oxynitrides. 5,70 Bond valence sum analyses have identified regions of under-bonded and over-bonded atom pairs that result from differences in covalency between the M –O and M –F interactions in refined average structures. 18,32 Lately, researchers are also using more sophisticated computational methods to study anion order in heteroanionic materials. Corradini et al. solved the complex structure of fluorinated anatase (TiO2 ) by devising a computational scheme that incorporates molecular dynamics (MD) and density functional theory (DFT) calculations. 48 DFT calculations have also been used to assess probable anion orders in the cathode material FeOF. 71 In this analysis, we assess the stability of the derivative structures with total energy density functional calculations. We perform cell optimizations (see SI for technical details) 18


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in 17 unique anion configurations, i.e., allowing the cell volume, cell shape and atomic positions to relax, for K2 NaTiOF5 (Fig. 5). It is important to note that although prior experimental measurements have demonstrated that K2 NaTiOF5 displays anion disorder, 72 structural analyses of this kind can help provide salient insights regarding the interplay between structural distortions, anion order, composition and driving forces. Here, we observe that by simply changing the relative positions of oxygen and fluorine in the double perovskite lattice of K2 NaTiOF5 results in a large spread in the energetic landscape. After optimization, we find the primary distortion for all anion variants is the displacement of the anions and cations to optimize the cation–anion bond lengths based on covalency differences (no rotations). The lowest energy oxide-fluoride configuration for K2 NaTiOF5 is orthorhombic in space group P mc21 . This variant is closely followed by anion ordering belonging to the P 42 /m, P mn21 , P ma2, and Cm space groups which all fall within 5 meV/atom. Interesting, Fig. 5 also shows that despite these energetic similarities certain anion configurations are very unfavorable, with energies > 15 meV/atom above the P mc21 structure. We find that the highest energy structural variants of K2 NaTiOF5 result from oxide-fluoride orderings where the four potassium sites possess the most irregularity in the oxide and fluoride coordination; for example, space group P 21 /m has one [KF12 ], two [KO3 F9 ] and one [KOF11 ] polyhedral units. The strong energetic competition among the various 0 K oxide-fluoride orderings is consistent with the prior observation that K2 NaTiOF5 displays anion sublattice disorder. Furthermore, for non-zero temperatures the entropic contributions play an important role in order/disorder phenomena. In fact, detailed calorimetry experiments have shown that phase transitions in related oxyfluoride perovskites are frequently accompanied by characteristic changes in entropy. 22,24,73 It is observed that small changes in lattice (vibrational) entropy (∆ S ≈ 0.1R – 0.5R) correspond to displacive transitions, while ordering transitions exhibit entropy changes of at least an order-of-magnitude larger. 22,24,73 In light of the dominant nature of the configurational entropy, we can approximate its contribution by computing the

19



Na

Mo

O

_

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F

Γ4+

Γ4

_ Fm3m disordered

R3 ordered & distorted

R3m ordered

¯ Figure 6: Progression of Na3 MoO3 F3 from cubic (F m3m) to trigonal (R3) subject to the − polar ordering irrep Γ4 and the out-of-phase octahedral tilt a− a− a− displacive irrep (Γ+ 4 ). ideal ideal entropy of mixing ∆ Smixing . 71 In the case of K2 NaTiOF5 , a system where 1/6 of the ideal fluoride ions is substituted by oxide ions, the ∆ Smixing is determined as follows,

ideal ∆Smixing

= −kB

1 1 5 5 ln + ln 6 6 6 6

≈ 0.039

meV , anion K

(3)

where kB is the Boltzmann constant. Although the ideal solution model is likely to overestimate the contributions from configurational entropy, 71 it provides a simple first approximation to understanding the effect of temperature on accessible anion orderings in lieu of computing the complete finite temperature phase diagram.

Application of Workflow for Understanding Phase Transitions There is evidence that on cooling, double perovskite oxyfluorides may undergo successive reversible phase transitions from the F m¯3m parent structure. 74,75 These reversible transitions are purely displacive in nature, i.e., related to octahedral tilting and secondary cation displacements, but are not related to ordering of oxide and fluoride ions. 18,76 This suggests that local anion ordering in these three-dimensionally corner-connected double perovskites is set during synthesis and is likely irreversible. Now we explore the sequence of such experimentally observed order-disorder and displacive transitions. + VI VI Na3 MoO3 F3 belongs to the A+ = Mo, W) 2 B M O3 F3 (A, B = Na, K, Rb, Tl, Cs; M

20


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family of oxyfluoride materials. 26,77 Like other double perovskites these compounds exhibit the F m¯3m parent at high temperature. 78 Prior crystallographic studies have shown that although the exact ordered-anion patterns in the family were difficult to decipher, most of the compounds are susceptible to octahedral rotations and exhibit fac-type [M VI O3 F3 ]3− anionic units. 25,26 + VI The compound Na3 MoO3 F3 is an exception to the trends observed in the A+ 2 B M O3 F 3

family. Na3 MoO3 F3 exhibits fac-type octahedral ordering but crystallizes in space group R3. 26 The primary structural distortion in Na3 MoO3 F3 is the cooperative a− a− a− octahedral tilt pattern resulting in a structure analogous to LiNbO3 . 79 The stability of the anion-ordered rhombohedral structure in Na3 MoO3 F3 has been attributed to steric effects: 26 In the F m¯3m structure, the small 12-coordinate Na cations are significantly underbonded and a large a− a− a− rotation reduces the Na coordination to 6-fold, improving its bond valence. Recently, Fry et al. proved the latter to be a demonstrable design strategy by substituting Na with the similar sized Ag cation to synthesize two new ordered oxyfluorides, Na1.5 Ag1.5 MoO3 F3 and Na1.5 Ag1.5 WO3 F3 that are isostructural to Na3 MoO3 F3 . 80 Here, we will show that our combined computational approach captures the F m¯3m → R3 transition in Na3 MoO3 F3 . Based on our data mining analysis, we determined that 29 out of 250 [M O3 F3 ]n− derivative structures exist with purely fac-type ordering. Direct analysis on this reduced set reveals that only one structure possesses an assembly of fac anionic units consistent with the Na3 MoO3 F3 ground state structure. This anion order lowers the symmetry of the parent from F m¯3m to R3m (Fig. 6). From the perspective of mode crystallography, the distortion responsible for the anion ordering is the polar Γ− 4 irrep. Our DFT lattice dynamical calculations performed on the optimized R3m structure found three unstable modes in the R3m anion ordered structure with the most unstable phonons at the Γ point (see Supporting Information, Figure S2). The softest mode has frequency 129i cm−1 and transforms R3m to R3. The two other unstable modes are degenerate at Γ (93i cm−1 ) and reduce R3m to P 1. Indeed, we find that “freezing in” the R3 imaginary mode stabilizes the ground state structure of Na3 MoO3 F3 (Fig. 6).

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We further note that we were also able to obtain the ground state structure of Na3 MoO3 F3 starting from the a− a− a− rotationally distorted homoanionic R3c structure of LiNbO3 . Superimposing the “correct” ordering of oxide and fluoride ions onto the distorted structure also results in the R3 symmetry. Although this post-anion ordering achieves the ground state Na3 MoO3 F3 phase, we emphasize that such a transition sequence is inconsistent with experimental observations.

Discussion Currently, the search for anion control in heteroanionic materials is largely guided by heuristics amassed over decades of successful (and unsuccessful) experimentation. Here, we present a scheme based on multiple levels of theory and computation to systematically test these heuristics and formulate crystal-chemistry principles for realizing anion order in HAMs. Our heteroanionic materials derivative structure generator is scalable and easily applicable to most chemistries or crystal classes. The code automates the otherwise cumbersome process of manually generating large numbers of unique anion ordered structures and organizes them by space group symmetry. The derivative structures represent the possible set of arrangements adopted by neighboring M Ox Xy BBUs (X =N3− , S2− , F1− ). When the anions occupy crystallographically distinct structural sites, structures with higher degrees of long-range anion order have higher space group symmetry. Experimentally, this is the case in compounds such as KNaNbOF5 (space group 129) and Na3 MoO3 F3 (space group 146). Structures where anion order lowers the space group symmetry to monoclinic and triclinic more closely resemble BBUs arrangements in orientational (substitutional) disordered crystals. As a result, aggregating a wide range of structural configurations enables one to study the connection between chemistry (ionic charge, size, orbital filling, etc.) and anion order. Prior work has shown that structural distortions such as octahedral rotations can influence anion order. 5,25,26 In addition, it has been speculated that the coupling between structural

22


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distortions and anion order may be key to stabilizing desirable properties in HAMs such as ferroelectricity. 10 However, the interplay between anion order and structural distortions has yet to be fully explored. Interestingly, our analysis of Glazer tilt distortions coupled to a hypothetical A2 BM OF5 lattice with P 4/nmm symmetry revealed that one could drive a centrosymmetric HAMs to possess a net dipole though a hybrid improper mechanism. This result provides evidence that coupling between anion order and distortions can strongly influence the properties of HAMs, but more importantly it provides an actionable method to circumvent the tendency HAMs to crystallize in centrosymmetric space groups. Data mining/analysis when used in tandem with the derivative structures database is a powerful tool to gain structural insights. Notably, our results suggest that local anion configurations (cis, trans, etc) within polyhedral units are linked to structural distortions that are strongly tied to cation chemistry such as polar (d0 transition metals) or Jahn-Teller (Mn3+ ) distortions. Moreover, we show that the local point group symmetry of the anion ordered BBU have space group symmetry directing properties and we catalog the allowed symmetries for cis, trans, fac, mer, and mixed isomers. Although fully (long-range) anion ordered heteroanionic materials may be difficult to achieve in experiment, we note that these structural trends have implications beyond hypothetically ordered structures. Many heteranionic materials, such as oxynitride perovskites can possess some degree of correlated anionic disorder. 5,10 We contend that structures with mixed-anion configurations may prove useful in the refinement of such structures. Moreover, the investigation of anion ordering principles using mode crystallography enables one to understand the origins of known anion ordered compounds and more intelligently select chemistries for new HAMs with functional properties. From our investigations of the chemistries KNa2 TiOF5 and Na3 MoO3 F3 , we showed that our combined computational approach effectively captures anion ordering phenomena. However, investigating anion order-disorder phenomenon from first principles is a difficult task. Generally speaking, anion order/disorder at finite temperature is a consequence of competing

23



energetic scales arising from enthalpic and entropic contributions. Therefore, the inclusion of entropic contributions are arguably critical to elucidating the physics governing anion order in HAMs. In this work as a first approximation, we include the largest entropic contribution (configurational) using the entropy of ideal mixing. The simple ideal mixing method likely overestimates the configurational entropy because it models complete site disorder and ignores the local order present in the heteroleptic units. However, this approximation enables one to screen for accessible structural variants. A more robust description of entropy towards the generation of finite temperature phase diagrams for heteroanionic materials may treat the anion ordered derivative structures as configurational microstates of a heteranionic system and then may include other entropic effects, for example lattice degrees of freedom using a quasi-harmonic approximation 81 within a DFT framework.

Conclusion Here we demonstrate a computational approach based on experimentally observed trends in heteroanionic materials which can be used to systematically study the structure-property relationships in with a wide range of chemistries and structure types. We successfully showed that the structural information derived from this approach can be used to understand phase transitions in known compounds and unlock hidden behavior such as hybrid improper ferroelectricity. Further work is needed to establish the full influence of anion order on physical properties, e.g., electronic structure changes, but we are confident that this approach will prove useful in understanding and building more quantitative anion ordering principles in HAMs. In fact, achieving active control of anion order could enable material scientists to reimagine materials platforms without inversion symmetry for technologies, such as nonlinear frequency mixing and piezoelectrics.

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Acknowledgement N.C. and J.M.R. acknowledge NSF (Award No. DMR-1454688) for financial support and thank K.R. Poeppelmeier for insightful discussions. R. Saballos was supported by the National Science Foundation’s MRSEC program (DMR-1720139) at the Materials Research Center of Northwestern University. DFT calculations were performed on the high-performance computing facilities available at the Center for Nanoscale Materials (CARBON Cluster) at Argonne National Laboratory, supported by the U.S. DOE, Office of Basic Energy Sciences (BES), DE-AC02-06CH11357.

Supporting Information Available Supplementary information is available in the online version of the paper: SOCCR algorithm performance metrics, anion ordered double perovskite derivative structure files, ordering statistics for A2 BM O2 F4 and A2 BM O3 F3 structures, and calculated phonons for Na3 MoO3 F3 with R3m symmetry. This material is available free of charge via the Internet at http://pubs.acs.org.

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TOC Entry MOF5 trans-MO2F4

cis-MO2F4

MX6

fac-MO3F3

mer-MO3F3

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Structural Diversity from Anion Order in Heteroanionic Materials

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